Suppose $A \in [a], B \in [b]$, and write $A = mp+a,B = np+b$ for some $m,n \in \mathbb{Z}$. Then, the statement that $[a][b] = [0]$ simply means that $p$ divides $AB = nmp + npa + mpb + ab$. The first 3 terms are clearly multiples of $p$ so then the last term is required to be a multiple of $p$ as well which leads to the requirement that $p$ divides $ab$.

The conclusion that $[a]$ or $[b]$ equals $[0]$ means that either $a$ or $b$ is a multiple of $p$. Taken together, the claim is that if $p$ divides $ab$, then $p$ divides either $a$ or $b$ as was pointed out by cjem.